Pressure at a point

In theorems 7.2.6, 7.2.7 and the remark afterwards, we considered the topological pressure on a point

z. Here, we prove the formulae that we quoted forP_{z}(ϕ),CP{z}(ϕ)and CP{z}(ϕ).

Theorem 7.6.1. LetX be a compact metric space, f :X7→X andz be an arbitrary point. Then

P_{z}(ϕ) =CP{z}(ϕ) = lim inf_n→∞ 1 n n_X−1 i=0 ϕ(fi(z)), CP_{z}(ϕ) = lim sup n→∞ 1 n n_X−1 i=0 ϕ(fi(z)).

Remark 7.6.1. It follows from theorem 7.6.1 and the ergodic theorem that for any invariant measure

µ, there is a set of full measure so thatP{z}(ϕ) =CP{z}(ϕ) =CP{z}(ϕ). Ifµis ergodic, this value isR ϕdµ.

Remark 7.6.2. If z is a point for which the Birkhoff average of ϕ does not exist, then P{z}(ϕ) =

CP_{z}(ϕ)< CP_{z}(ϕ).

The theorem is a consequence of the lemmas that follow and the relationPZ(ϕ)≤CPZ(ϕ)≤ CPZ(ϕ) for any Borel set Z ⊂X (formula (11.9) of [Pes]).

Lemma 7.6.1. Let (X, d) be a compact metric space, ϕ : X 7→ R _{a continuous function, and} z∈X. Then P{z}(ϕ)≥lim inf_n→∞ 1 n n_X−1 i=0 ϕ(fi(z)).

Proof. We work directly with our usual definition of Pesin and Pistskel topological pressure (see

§2.1.1). Without loss of generality, it suffices to consider covers of {z} by a single setBn(x, ε). Fix ε >0,N ∈N_{and 0< δ <} 1 2. Choose α satisfying α <lim inf n→∞ 1 n n_X−1 i=0 ϕ(fi(z))−Var(ϕ, ε)−δ. (7.5) AssumeN was chosen sufficiently large so that form≥N,

1 n n_X−1 i=0 ϕ(fi(z))≥lim inf n→∞ 1 n n_X−1 i=0 ϕ(fi(z))−δ. (7.6) ChooseΓ ={Bm(x, ε)} such thatz∈Bm(x, ε),m≥N and

|Q({z}, α,Γ, ϕ)−M({z}, α, ε, N, ϕ)| ≤δ.

We can prove thatPm_k=0−1ϕ(fk_{(z))−mVar(ϕ, ε)−mα >0, which follows from (7.5) and (7.6). It}

follows that M({z}, α, ε, N, ϕ) ≥ exp ( −αm+ sup y∈Bm(x,ε) m_X−1 k=0 ϕ(fk(y)) ) −δ ≥ exp ( −αm+ m_X−1 k=0 ϕ(fk(z))−mVar(ϕ, ε) ) −δ ≥ 1−δ≥ 1 2.

SoM({z}, α, ε, N, ϕ)>0 and henceP{z}(ϕ, ε)≥α. It follows that

P{z}(ϕ, ε)≥lim inf_n→∞ 1 n n_X−1 i=0 ϕ(fi(z))−Var(ϕ, ε)−δ.

On taking the limitε→0and noting that δ was arbitrary, we obtain the desired result.

Lemma 7.6.2. CP_{z}(ϕ) = lim infn→∞_n1 Pin=0−1ϕ(fi(z)).

Proof. It follows from the definition ofCPZ(ϕ) that CP_{z}(ϕ) = lim ε→0lim infn→∞ 1 nlog x:z∈infBn(x,ε) exp (_n−1 X i=0 ϕ(fi(x)) )! .

For a fixedεandBn(x, ε) which containsz, n_X−1 i=0 ϕ(fi(x))≤ n_X−1 i=0 ϕ(fi(z)) +nγ(ε).

It follows that CP{z}(ϕ)≤lim ε→0lim infn→∞ 1 n{ n_X−1 i=0 ϕ(fi(z)) +γ(ε)}.

We obtainCP_{z}(ϕ)≤lim infn→∞_n1Pni=0−1ϕ(fi(z))and we can prove the reverse inequality in the

same way.

Lemma 7.6.3. CP{z}(ϕ) = lim supn→∞ n1 Pn−1

i=0 ϕ(fi(z)).

Proof. It follows from the definition ofCPZ(ϕ) that CP_{z}(ϕ) = lim_ε→0lim sup

n→∞ 1 nlog x:z∈infBn(x,ε) exp (_n−1 X i=0 ϕ(fi(x)) )! .

Future directions

We mention some questions of further interest which relate to the contents of the thesis.

There are some very interesting questions surrounding the almost specification property (see chapter 6). Many interesting results which are known for maps with the specification property should generalise to the class of maps with almost specification. For example, Bowen’s results concerning uniqueness of equilibrium states for maps with the specification property [Bow5] should carry over to this more general setting.

Another obvious avenue of investigation is to see which other maps have almost specification. We have some ideas about this problem in the context of piecewise monotonic interval maps and for certain examples of shift spaces. However, this project has not come to fruition in time to be included in the thesis.

In §6.5.3, we found subshifts of finite type within the β-shift with entropy arbitrarily close to logβ. This suggests the investigation of a ‘horseshoe’ method of proof for results about the topological entropy of the irregular set. More precisely, we could study the class of systems(X, f)

which contain subsystems(Y, fn_{)which are topological factors of shifts of finite type, whereY ⊂X}

is compact andn∈N_{. We call(Y, f}n_{) a horseshoe for (X, f). If the entropy of the horseshoe can}

be chosen to approximate that of the whole space arbitrarily well, then it should suffice to study the intersection of the irregular set with the horseshoe. We note that systems with specification do not necessarily contain any horseshoes, so this approach would not recover our current results. Also, theorems on the existence of horseshoes typically require smoothness of the system (see, for example, theorem S.5.9 of [KH]), whereas our current approach is a topological approach to a topological question. However, we do note that examples exist that do not have specification but where a ‘horseshoe’ approach could yield results. For example, a continuous interval map which is not mixing contains horseshoes but does not have specification (see corollary 15.2.10 of [KH]). Thus, the ‘horseshoe’ approach certainly has merit.

An idea from the thesis which we hope will prove useful is to only ask for specification to hold on an interesting invariant non-compact subsetX′ ⊂X (see definition 2.2.3). The idea could have applications for the study of non-uniformly hyperbolic systems. The interesting invariant set

X′ to which we allude is the set of points which return infinitely often to a set on which the map is uniformly hyperbolic. We hope to pursue this in the future. A particular avenue of interest for this is the Rauzy-Veech map and Teichm¨uller flow [Buf], which are related systems of great current interest arising from geometry.

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